The social planner maximizes aggregate human capital subject to a budget constraint
B = 2H, so that the per capita budget is 2 units of investment. We draw H children from
the initial distribution F (θ1,h), and solve the problem of how to allocate finite resources 2H
to maximize the average education of the cohort. Formally, the social planner maximizes
aggregate schooling
1H
max S= H∑S (θc,3,h, θN,3,h, ∏h),
h=1
subject to the aggregate budget constraint,
H
X (I1,h + I2,h) = 2H, (4.5)
h=1
the technology constraint,
θk,t+1,h = fk,t (θC,t,h , θN,t,h , θC,P,h , θN,P,h , πh) for k ∈ {C, N} and t ∈ {1, 2},
and the initial endowments of the child and her family. We assume no discounting. Solving
this problem, we obtain optimal early and late investments, I1,h and I2,h , respectively, for
each child h. An analogous social planning problem is used to minimize crime.
Figures 2 (for the child’s personal endowments) and 3 (for maternal endowments) show
the profiles of early (left-hand side graph) and late (right-hand side graph) investment as
a function of child and maternal endowments (lighter shading corresponds to higher values
of investment). Endowments are measured in units of standard deviations from the means.
In each figure, the endowments not plotted are fixed at sample mean values. The optimal
policy is to invest relatively more in the disadvantaged compared to the advantaged in the
early years. Moon (2010) shows that, in actuality, society and family together invest much
more in the early years of the advantaged compared to the disadvantaged. The decline in
investment by level of advantage is dramatic for early investment. Second period investment
profiles are much flatter and slightly favor relatively more investment in more advantaged
children. A similar profile emerges for investments to reduce aggregate crime, which for the
sake of brevity, we do not display.
Figures 4 and 5 reveal that the ratio of optimal early-to-late investment as a function of
the child’s personal endowments declines with advantage whether the social planner seeks to
maximize educational attainment (left hand side) or to minimize aggregate crime (right hand
side). A somewhat similar pattern emerges for the optimal ratio of early-to-late investment
varying child quality is considerably more complicated because of the high dimensionality of the state space.
We leave this for another occasion.
33